Do we declare the codomain of a function from the beginning, or do we determine it after defining the domain and operations?

Question

If we have a function, would we assume/declare the codomain from the beginning, along with when we define the domain and operations, or would we determine the codomain afterwards, like after we have already chosen the domain and valid operations? Also, if we had equations or formulas or definitions, then would we assume/declare the number system of the "output" variable from the beginning (at the same time when we define the domain and operations for the equation/formula) (meaning that the "output" variable's number system is something we have to define beforehand as a part of the function/equation/formula and it cannot be determined afterwards)? Or would we find/determine the number system for the "output" variable after we have already defined the number system for the "input" variable and its operations, and after we evaluate that part of the equation/formula (on one side of the equals sign), so we know what the "output" variable will be equal to? Also, from what I understand (please correct me if I am wrong), the codomain basically states/defines the general/overall number system that the output will be in (like the real numbers, complex numbers, etc.) (so we know that we can treat the equation/equality as a valid real number or complex number), and the range is the specific numbers that the output can actually be (which we find later as a specific subset of the codomain), right?

For example, if we have a function $f(x)=\sqrt{x}$, then I think that this function, just alone, would not be properly defined since its domain and operations aren't declared. But if we declare that the domain for $f(x)$ is $\Bbb R^+$ and the operations also take place in $\Bbb R$, then which of the following is correct (1 or 2)?

1. We also have to declare beforehand that the codomain of $f(x)$ is $\Bbb R$ as a definition of the function, along with its domain and operations. So basically, we state/declare/assume that $f(x)$ is a real-valued function beforehand. This way, we ensure that we have an equality $(f(x)=\sqrt{x})$ between real numbers on the LHS and RHS.

2. We determine afterwards that the codomain of $f(x)$ must be in the real numbers after we declare that the domain of $f(x)$ is the real numbers and the operations are also in the real numbers. So basically, we determine that $f(x)$ is a real-valued function after evaluating $\sqrt{x}$.

Also, how would this apply to other mathematical equalities, like equations or formulas? Because I know that when we solve or rearrange equations for a variable, then we must assume/declare that the variable, equation, and operations take place in a specific number system for this to be valid (I asked this question before here, here, and here) (ex. If we have $x^2=-4$, we must declare beforehand that our variable $(x)$, operations, and equation take place in the complex number system to get a valid answer of $x=±2i$). So, for example, let's say we have $V=IR$ (ohm's law) or $A(r)=\pi r^2$ (area of a circle).

1. So, for $V=IR$, do we determine that $V$ must be in the real numbers after we declare that $I$ and $R$ are reals and our operation of multiplication is taking place in the real numbers? Or do we already have to assume beforehand that $V$ must be a real number, along with us assuming that $I$ and $R$ are real numbers (so then this way, we already declared that $V$, $I$, and $R$ are real numbers, and then since we know that $IR$ must also be a real number (since we are doing the multiplication operation in the real numbers), we know that our equality between $V$ and $IR$ will be valid in the real numbers, so it's a valid equation that can be used as a formula (to find the value of an unknown variable ($V$, $I$, or $R$) in the real numbers))?

2. And for $A(r)=\pi r^2$, when we assume that $r \in \Bbb{R}^+$ as its domain, does that tell us/prove that the area is a real number as well, or do we have to assume/declare beforehand that $A(r)$ will also be a real number to use the formula (so that we know we have a true equality between real numbers, and we can use it as a formula for the area of a circle)?

Also, how would this apply to mathematical definitions? Like, would this same process of defining the codomain of a function $f(x)$ also apply to mathematical definitions (like the definition of a derivative or integral)? Like, let's say we define the derivative to be the limit of the difference quotient, or we define the integral to be the limit of a Riemann sum. So then, from the beginning, would we have to assume/declare that the derivative ($\frac{\mathrm{d}y}{\mathrm{d}x}$) or integral ($\int_{a}^{b} f(x) \, \mathrm{d}x$) are a part of the real numbers (meaning we just defined the codomain, where $\frac{\mathrm{d}y}{\mathrm{d}x}$ or $\int_{a}^{b} f(x) \, \mathrm{d}x$ would represent the function $f(x)$), along with also defining from the beginning that the input and operations in the limits/sums are in the real numbers (meaning we just defined the domain and operations, where the limit of the different quotient (for defining $\frac{\mathrm{d}y}{\mathrm{d}x}$) or the limit of a Riemann sum (for defining $\int_{a}^{b} f(x) \, \mathrm{d}x$) would be the operations defining the function $f(x)$), to fully define the derivative or integral?

So, overall, I would like to know more about these assumptions that are required when defining the equality in a function, formula, equation, or definition. Any help regarding these assumptions for functions/equations/formulas/definitions would be greatly appreciated! Thank you! (Sorry for the long question, please let me know if any clarifications are needed.)

Answer 1

I find this a confusing question because it is hard to tell whether it is talking about the way mathematics is or about the way we do mathematics.

When you do mathematics, there are likely to be a lot of false starts and dead ends. You think you have a useful idea but it doesn't work out well. You start trying one thing but then you think of something else and decide to do that instead.

You might start out thinking you're working on a problem in real analysis but later decide that this would be better as a problem in complex analysis.

So it doesn't seem practical to make all the choices you're going to need to make "at the beginning." We need some flexibility in how we do mathematics.

But when you finally are ready to write a proof of something, you can't have the codomain of a function mysteriously change during the proof. In that context you need to complete the definitions of things when you define them; the definition of an object is the "beginning" from which all the necessary assumptions and declarations of that definition should be in place. You don't want to be saying "I forgot to mention" explicitly or implicitly.

Answer 2

Edit: the fact that the codomain is not properly taught in high school maths has bothered me for a long time. Here is an old question of mine on MathsEducation.SE.

This is a complement of @Michael_1812's excellent answer on what a function is.

Before starting, note that there may be different definitions of functions depending on the area of maths you are. I will focus on the precalculus/calculus.

First of all, it seems that there is a misunderstanding about what a function is. The definition of a function is:

Let $A$ and $B$ be two sets. A function $f$ from $A$ to $B$ is a relation that associates t each element of $A$ a unique element of $B$. $A$ is called the domain and $B$ the codomain of $f$. We write $f:A \to B$ to show that the domain is $A$ and the codomain is $B$.

So a function is more than a formula or expression, it is a way to associate the elements of one set to another. The intuitive way of describing a function as a "machine" that takes some input $x$ from $A$ and transforms it into an output $f(x)$ in $B$ is I think how all mathematicians think about functions.

And you can see from the definition that the domain and the codomain are integral parts of the definition. If you change the domain, you change the definition. For example, the functions $f:[0,\infty)\to\mathbb R$ and $g:\mathbb R\to\mathbb R$ defined by $f(x)=x^2$ and $g(x)=x^2$ are different functions. Simply because they have different domains. In particular, $f(-1)$ is undefined. More importantly, they have different properties, $f$ being increasing, hence having an inverse; while $g$ is not.

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In Precalculus, the domain $A$ and the codomain $B$ will be subsets of $\mathbb R$, bare a few examples. But they don't need to, and it is important, that many things in mathematics can be described as a function. For example, the area of a triangle can be described as a function, from the set $T$ of all triangles in the plane (the domain) to the set of all real numbers.

The fact that in Precalculus, the domain and the range of a function will always be subsets of $\mathbb R$ lead people to adopt a shorter notation like simply writing $f(x)$ as a formula, for example, $f(x)=x^2$, or an expression like $y=x^2$. When this is done, one has to keep two conventions in mind:

1. if the domain is not explicitly given, the domain is the largest subset of $\mathbb R$ such that the expression makes sense. For example, the domain of $y=x^2$ is $\mathbb R$ while the domain of $y=\frac{1}{\sqrt{x}}$ is $(0,\infty)$. Some books call this the implicit domain
2. If the codomain is not explicitly given, the codomain is the range. This allows to write things like "a function has an inverse if and only if it is one-to-one".

Answer 3

"which of the following is correct (1 or 2)?"

Correct is No 1.

We define a function, or mapping, as $$ f:~X\longrightarrow Y\;, $$ the domain $X$ and codomain $Y$ being some sets. This being said, we do not imply that each element of $Y$ has a preimage.

If the mapping is onto, we say that the image of $X$ or, the same, the range of $f$ coincides with the codomain $Y$.

If the mapping is into, the image of $X$ (the range of values of $f$) is residing in $Y$ but is not identical to it.

This picture from Wikipedia illustrates the concept. The codomain is blue, the image (range) is yellow.